In his influential book `The Principles of Mathematics Revisited', Jaakko Hintikka discusses independence-friendly first-order logic (IF-FOL for short) and how drastically different the situation in the philosophy of mathematics would be if classical logic were replaced by IF-FOL. Inspired by Hintikka's ideas on constructivism, we will 'effectivize' the game-theoretic semantics (abbreviated GTS)for IF-FOL, but in a somewhat different way than he did in the book.

Here is how, in a nutshell: first we show that Nelson's realizability interpretation - which extends the famous Kleene's realizability interpretation by adding `strong negation' - restricted to the implication-free first-order formulas can be viewed as an effective version of GTS for first-order logic. Then we propose a realizability interpretation for IF-FOL, inspired by the so-called trump semantics which was discovered by Wilfrid Hodges, and show that this trump realizability interpretation can be viewed as an effective version of GTS for IF-FOL.

Finally we prove that the trump realizability interpretation for IF-FOL appropriately generalises Nelson's restricted realizability interpretation for the implication-free firstorder formulas. In the light of these results, we shall review Hintikka's constuctivistic ideas, and also touch on the problem of adding implications to IF-FOL in a way consistent with constructivism.